\(\int e^{-18-2 e^6-2 e^3 (2-2 x)+6 x-2 x^2} (1-e^{18+2 e^6+2 e^3 (2-2 x)-6 x+2 x^2}+6 x+4 e^3 x-4 x^2) \, dx\) [852]

Optimal result

Integrand size = 73, antiderivative size = 26 \[ \int e^{-18-2 e^6-2 e^3 (2-2 x)+6 x-2 x^2} \left (1-e^{18+2 e^6+2 e^3 (2-2 x)-6 x+2 x^2}+6 x+4 e^3 x-4 x^2\right ) \, dx=5-x+e^{-16-2 \left (1+e^3-x\right )^2+2 x} x \] Output:

x/exp((exp(3)-x+1)^2+8-x)^2+5-x
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int e^{-18-2 e^6-2 e^3 (2-2 x)+6 x-2 x^2} \left (1-e^{18+2 e^6+2 e^3 (2-2 x)-6 x+2 x^2}+6 x+4 e^3 x-4 x^2\right ) \, dx=\left (-1+e^{-2 \left (9+e^6-2 e^3 (-1+x)-3 x+x^2\right )}\right ) x \] Input:

Integrate[E^(-18 - 2*E^6 - 2*E^3*(2 - 2*x) + 6*x - 2*x^2)*(1 - E^(18 + 2*E 
^6 + 2*E^3*(2 - 2*x) - 6*x + 2*x^2) + 6*x + 4*E^3*x - 4*x^2),x]
 

Output:

(-1 + E^(-2*(9 + E^6 - 2*E^3*(-1 + x) - 3*x + x^2)))*x
 

Rubi [A] (verified)

Time = 1.62 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.055, Rules used = {6, 7292, 7293, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \left (-4 x^2-e^{2 x^2-6 x+2 e^3 (2-2 x)+2 e^6+18}+4 e^3 x+6 x+1\right ) \exp \left (-2 x^2+6 x-2 e^3 (2-2 x)-2 e^6-18\right ) \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \left (-4 x^2-e^{2 x^2-6 x+2 e^3 (2-2 x)+2 e^6+18}+\left (6+4 e^3\right ) x+1\right ) \exp \left (-2 x^2+6 x-2 e^3 (2-2 x)-2 e^6-18\right )dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \left (-4 x^2-e^{2 x^2-6 x+2 e^3 (2-2 x)+2 e^6+18}+\left (6+4 e^3\right ) x+1\right ) \exp \left (-2 x^2+2 \left (3+2 e^3\right ) x-2 \left (9+2 e^3+e^6\right )\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-4 x^2 \exp \left (-2 x^2+2 \left (3+2 e^3\right ) x-2 \left (9+2 e^3+e^6\right )\right )+2 \left (3+2 e^3\right ) x \exp \left (-2 x^2+2 \left (3+2 e^3\right ) x-2 \left (9+2 e^3+e^6\right )\right )+\exp \left (-2 x^2+2 \left (3+2 e^3\right ) x-2 \left (9+2 e^3+e^6\right )\right )-1\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle x \exp \left (-2 x^2+2 \left (3+2 e^3\right ) x-2 \left (9+2 e^3+e^6\right )\right )-x\)

Input:

Int[E^(-18 - 2*E^6 - 2*E^3*(2 - 2*x) + 6*x - 2*x^2)*(1 - E^(18 + 2*E^6 + 2 
*E^3*(2 - 2*x) - 6*x + 2*x^2) + 6*x + 4*E^3*x - 4*x^2),x]
 

Output:

-x + E^(-2*(9 + 2*E^3 + E^6) + 2*(3 + 2*E^3)*x - 2*x^2)*x
 

Defintions of rubi rules used

rule 6

Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v 
+ (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] &&  !FreeQ[Fx, x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19

method	result	size
risch	\(-x +x \,{\mathrm e}^{4 x \,{\mathrm e}^{3}-2 x^{2}-4 \,{\mathrm e}^{3}-2 \,{\mathrm e}^{6}+6 x -18}\)	\(31\)
parallelrisch	\(-\left ({\mathrm e}^{2 \,{\mathrm e}^{6}+2 \left (2-2 x \right ) {\mathrm e}^{3}+2 x^{2}-6 x +18} x -x \right ) {\mathrm e}^{-2 \,{\mathrm e}^{6}-2 \left (2-2 x \right ) {\mathrm e}^{3}-2 x^{2}+6 x -18}\)	\(55\)
default	\(-x +\frac {{\mathrm e}^{-2 \,{\mathrm e}^{6}} {\mathrm e}^{-4 \,{\mathrm e}^{3}} {\mathrm e}^{-18} \sqrt {\pi }\, {\mathrm e}^{\frac {\left (6+4 \,{\mathrm e}^{3}\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, x -\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {2}}{4}\right )}{4}+6 \,{\mathrm e}^{-2 \,{\mathrm e}^{6}} {\mathrm e}^{-4 \,{\mathrm e}^{3}} {\mathrm e}^{-18} \left (-\frac {{\mathrm e}^{-2 x^{2}+\left (6+4 \,{\mathrm e}^{3}\right ) x}}{4}+\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {\pi }\, {\mathrm e}^{\frac {\left (6+4 \,{\mathrm e}^{3}\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, x -\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {2}}{4}\right )}{16}\right )-4 \,{\mathrm e}^{-2 \,{\mathrm e}^{6}} {\mathrm e}^{-4 \,{\mathrm e}^{3}} {\mathrm e}^{-18} \left (-\frac {x \,{\mathrm e}^{-2 x^{2}+\left (6+4 \,{\mathrm e}^{3}\right ) x}}{4}+\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \left (-\frac {{\mathrm e}^{-2 x^{2}+\left (6+4 \,{\mathrm e}^{3}\right ) x}}{4}+\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {\pi }\, {\mathrm e}^{\frac {\left (6+4 \,{\mathrm e}^{3}\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, x -\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {2}}{4}\right )}{16}\right )}{4}+\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {\left (6+4 \,{\mathrm e}^{3}\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, x -\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {2}}{4}\right )}{16}\right )+4 \,{\mathrm e}^{-2 \,{\mathrm e}^{6}} {\mathrm e}^{-4 \,{\mathrm e}^{3}} {\mathrm e}^{-18} {\mathrm e}^{3} \left (-\frac {{\mathrm e}^{-2 x^{2}+\left (6+4 \,{\mathrm e}^{3}\right ) x}}{4}+\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {\pi }\, {\mathrm e}^{\frac {\left (6+4 \,{\mathrm e}^{3}\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, x -\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {2}}{4}\right )}{16}\right )\)	\(361\)
parts	\(-x +\frac {{\mathrm e}^{-2 \,{\mathrm e}^{6}} {\mathrm e}^{-4 \,{\mathrm e}^{3}} {\mathrm e}^{-18} \sqrt {\pi }\, {\mathrm e}^{\frac {\left (6+4 \,{\mathrm e}^{3}\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, x -\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {2}}{4}\right )}{4}+6 \,{\mathrm e}^{-2 \,{\mathrm e}^{6}} {\mathrm e}^{-4 \,{\mathrm e}^{3}} {\mathrm e}^{-18} \left (-\frac {{\mathrm e}^{-2 x^{2}+\left (6+4 \,{\mathrm e}^{3}\right ) x}}{4}+\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {\pi }\, {\mathrm e}^{\frac {\left (6+4 \,{\mathrm e}^{3}\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, x -\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {2}}{4}\right )}{16}\right )-4 \,{\mathrm e}^{-2 \,{\mathrm e}^{6}} {\mathrm e}^{-4 \,{\mathrm e}^{3}} {\mathrm e}^{-18} \left (-\frac {x \,{\mathrm e}^{-2 x^{2}+\left (6+4 \,{\mathrm e}^{3}\right ) x}}{4}+\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \left (-\frac {{\mathrm e}^{-2 x^{2}+\left (6+4 \,{\mathrm e}^{3}\right ) x}}{4}+\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {\pi }\, {\mathrm e}^{\frac {\left (6+4 \,{\mathrm e}^{3}\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, x -\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {2}}{4}\right )}{16}\right )}{4}+\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {\left (6+4 \,{\mathrm e}^{3}\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, x -\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {2}}{4}\right )}{16}\right )+4 \,{\mathrm e}^{-2 \,{\mathrm e}^{6}} {\mathrm e}^{-4 \,{\mathrm e}^{3}} {\mathrm e}^{-18} {\mathrm e}^{3} \left (-\frac {{\mathrm e}^{-2 x^{2}+\left (6+4 \,{\mathrm e}^{3}\right ) x}}{4}+\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {\pi }\, {\mathrm e}^{\frac {\left (6+4 \,{\mathrm e}^{3}\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, x -\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {2}}{4}\right )}{16}\right )\)	\(361\)

Input:

int((-exp(exp(3)^2+(2-2*x)*exp(3)+x^2-3*x+9)^2+4*x*exp(3)-4*x^2+6*x+1)/exp 
(exp(3)^2+(2-2*x)*exp(3)+x^2-3*x+9)^2,x,method=_RETURNVERBOSE)
 

Output:

-x+x*exp(4*x*exp(3)-2*x^2-4*exp(3)-2*exp(6)+6*x-18)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (24) = 48\).

Time = 0.09 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.00 \[ \int e^{-18-2 e^6-2 e^3 (2-2 x)+6 x-2 x^2} \left (1-e^{18+2 e^6+2 e^3 (2-2 x)-6 x+2 x^2}+6 x+4 e^3 x-4 x^2\right ) \, dx=-{\left (x e^{\left (2 \, x^{2} - 4 \, {\left (x - 1\right )} e^{3} - 6 \, x + 2 \, e^{6} + 18\right )} - x\right )} e^{\left (-2 \, x^{2} + 4 \, {\left (x - 1\right )} e^{3} + 6 \, x - 2 \, e^{6} - 18\right )} \] Input:

integrate((-exp(exp(3)^2+(2-2*x)*exp(3)+x^2-3*x+9)^2+4*x*exp(3)-4*x^2+6*x+ 
1)/exp(exp(3)^2+(2-2*x)*exp(3)+x^2-3*x+9)^2,x, algorithm="fricas")
 

Output:

-(x*e^(2*x^2 - 4*(x - 1)*e^3 - 6*x + 2*e^6 + 18) - x)*e^(-2*x^2 + 4*(x - 1 
)*e^3 + 6*x - 2*e^6 - 18)
 

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int e^{-18-2 e^6-2 e^3 (2-2 x)+6 x-2 x^2} \left (1-e^{18+2 e^6+2 e^3 (2-2 x)-6 x+2 x^2}+6 x+4 e^3 x-4 x^2\right ) \, dx=x e^{- 2 x^{2} + 6 x - 2 \cdot \left (2 - 2 x\right ) e^{3} - 2 e^{6} - 18} - x \] Input:

integrate((-exp(exp(3)**2+(2-2*x)*exp(3)+x**2-3*x+9)**2+4*x*exp(3)-4*x**2+ 
6*x+1)/exp(exp(3)**2+(2-2*x)*exp(3)+x**2-3*x+9)**2,x)
 

Output:

x*exp(-2*x**2 + 6*x - 2*(2 - 2*x)*exp(3) - 2*exp(6) - 18) - x
 

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.24 (sec) , antiderivative size = 397, normalized size of antiderivative = 15.27 \[ \int e^{-18-2 e^6-2 e^3 (2-2 x)+6 x-2 x^2} \left (1-e^{18+2 e^6+2 e^3 (2-2 x)-6 x+2 x^2}+6 x+4 e^3 x-4 x^2\right ) \, dx=\frac {1}{4} \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} x - \frac {1}{2} \, \sqrt {2} {\left (2 \, e^{3} + 3\right )}\right ) e^{\left (\frac {1}{2} \, {\left (2 \, e^{3} + 3\right )}^{2} - 2 \, e^{6} - 4 \, e^{3} - 18\right )} - \frac {1}{2} i \, \sqrt {2} {\left (\frac {i \, \sqrt {\pi } {\left (2 \, x - 2 \, e^{3} - 3\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {{\left (2 \, x - 2 \, e^{3} - 3\right )}^{2}}\right ) - 1\right )} {\left (2 \, e^{3} + 3\right )}}{\sqrt {{\left (2 \, x - 2 \, e^{3} - 3\right )}^{2}}} - i \, \sqrt {2} e^{\left (-\frac {1}{2} \, {\left (2 \, x - 2 \, e^{3} - 3\right )}^{2}\right )}\right )} e^{\left (\frac {1}{2} \, {\left (2 \, e^{3} + 3\right )}^{2} - 2 \, e^{6} - 4 \, e^{3} - 15\right )} + \frac {1}{4} i \, \sqrt {2} {\left (\frac {i \, \sqrt {\pi } {\left (2 \, x - 2 \, e^{3} - 3\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {{\left (2 \, x - 2 \, e^{3} - 3\right )}^{2}}\right ) - 1\right )} {\left (2 \, e^{3} + 3\right )}^{2}}{\sqrt {{\left (2 \, x - 2 \, e^{3} - 3\right )}^{2}}} - 2 i \, \sqrt {2} {\left (2 \, e^{3} + 3\right )} e^{\left (-\frac {1}{2} \, {\left (2 \, x - 2 \, e^{3} - 3\right )}^{2}\right )} - \frac {2 i \, {\left (2 \, x - 2 \, e^{3} - 3\right )}^{3} \Gamma \left (\frac {3}{2}, \frac {1}{2} \, {\left (2 \, x - 2 \, e^{3} - 3\right )}^{2}\right )}{{\left ({\left (2 \, x - 2 \, e^{3} - 3\right )}^{2}\right )}^{\frac {3}{2}}}\right )} e^{\left (\frac {1}{2} \, {\left (2 \, e^{3} + 3\right )}^{2} - 2 \, e^{6} - 4 \, e^{3} - 18\right )} - \frac {3}{4} i \, \sqrt {2} {\left (\frac {i \, \sqrt {\pi } {\left (2 \, x - 2 \, e^{3} - 3\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {{\left (2 \, x - 2 \, e^{3} - 3\right )}^{2}}\right ) - 1\right )} {\left (2 \, e^{3} + 3\right )}}{\sqrt {{\left (2 \, x - 2 \, e^{3} - 3\right )}^{2}}} - i \, \sqrt {2} e^{\left (-\frac {1}{2} \, {\left (2 \, x - 2 \, e^{3} - 3\right )}^{2}\right )}\right )} e^{\left (\frac {1}{2} \, {\left (2 \, e^{3} + 3\right )}^{2} - 2 \, e^{6} - 4 \, e^{3} - 18\right )} - x \] Input:

integrate((-exp(exp(3)^2+(2-2*x)*exp(3)+x^2-3*x+9)^2+4*x*exp(3)-4*x^2+6*x+ 
1)/exp(exp(3)^2+(2-2*x)*exp(3)+x^2-3*x+9)^2,x, algorithm="maxima")
 

Output:

1/4*sqrt(2)*sqrt(pi)*erf(sqrt(2)*x - 1/2*sqrt(2)*(2*e^3 + 3))*e^(1/2*(2*e^ 
3 + 3)^2 - 2*e^6 - 4*e^3 - 18) - 1/2*I*sqrt(2)*(I*sqrt(pi)*(2*x - 2*e^3 - 
3)*(erf(sqrt(1/2)*sqrt((2*x - 2*e^3 - 3)^2)) - 1)*(2*e^3 + 3)/sqrt((2*x - 
2*e^3 - 3)^2) - I*sqrt(2)*e^(-1/2*(2*x - 2*e^3 - 3)^2))*e^(1/2*(2*e^3 + 3) 
^2 - 2*e^6 - 4*e^3 - 15) + 1/4*I*sqrt(2)*(I*sqrt(pi)*(2*x - 2*e^3 - 3)*(er 
f(sqrt(1/2)*sqrt((2*x - 2*e^3 - 3)^2)) - 1)*(2*e^3 + 3)^2/sqrt((2*x - 2*e^ 
3 - 3)^2) - 2*I*sqrt(2)*(2*e^3 + 3)*e^(-1/2*(2*x - 2*e^3 - 3)^2) - 2*I*(2* 
x - 2*e^3 - 3)^3*gamma(3/2, 1/2*(2*x - 2*e^3 - 3)^2)/((2*x - 2*e^3 - 3)^2) 
^(3/2))*e^(1/2*(2*e^3 + 3)^2 - 2*e^6 - 4*e^3 - 18) - 3/4*I*sqrt(2)*(I*sqrt 
(pi)*(2*x - 2*e^3 - 3)*(erf(sqrt(1/2)*sqrt((2*x - 2*e^3 - 3)^2)) - 1)*(2*e 
^3 + 3)/sqrt((2*x - 2*e^3 - 3)^2) - I*sqrt(2)*e^(-1/2*(2*x - 2*e^3 - 3)^2) 
)*e^(1/2*(2*e^3 + 3)^2 - 2*e^6 - 4*e^3 - 18) - x
 

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.14 (sec) , antiderivative size = 134, normalized size of antiderivative = 5.15 \[ \int e^{-18-2 e^6-2 e^3 (2-2 x)+6 x-2 x^2} \left (1-e^{18+2 e^6+2 e^3 (2-2 x)-6 x+2 x^2}+6 x+4 e^3 x-4 x^2\right ) \, dx=\frac {1}{2} \, \sqrt {2} \sqrt {\pi } {\left (2 \, e^{3} + 3\right )} \operatorname {erf}\left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - 2 \, e^{3} - 3\right )}\right ) e^{\left (2 \, e^{3} - \frac {21}{2}\right )} - \frac {1}{2} \, \sqrt {2} \sqrt {\pi } {\left (2 \, e^{6} + 3 \, e^{3}\right )} \operatorname {erf}\left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - 2 \, e^{3} - 3\right )}\right ) e^{\left (2 \, e^{3} - \frac {27}{2}\right )} + {\left (x + e^{3}\right )} e^{\left (-2 \, x^{2} + 4 \, x e^{3} + 6 \, x - 2 \, e^{6} - 4 \, e^{3} - 18\right )} - x - e^{\left (-2 \, x^{2} + 4 \, x e^{3} + 6 \, x - 2 \, e^{6} - 4 \, e^{3} - 15\right )} \] Input:

integrate((-exp(exp(3)^2+(2-2*x)*exp(3)+x^2-3*x+9)^2+4*x*exp(3)-4*x^2+6*x+ 
1)/exp(exp(3)^2+(2-2*x)*exp(3)+x^2-3*x+9)^2,x, algorithm="giac")
 

Output:

1/2*sqrt(2)*sqrt(pi)*(2*e^3 + 3)*erf(1/2*sqrt(2)*(2*x - 2*e^3 - 3))*e^(2*e 
^3 - 21/2) - 1/2*sqrt(2)*sqrt(pi)*(2*e^6 + 3*e^3)*erf(1/2*sqrt(2)*(2*x - 2 
*e^3 - 3))*e^(2*e^3 - 27/2) + (x + e^3)*e^(-2*x^2 + 4*x*e^3 + 6*x - 2*e^6 
- 4*e^3 - 18) - x - e^(-2*x^2 + 4*x*e^3 + 6*x - 2*e^6 - 4*e^3 - 15)
 

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int e^{-18-2 e^6-2 e^3 (2-2 x)+6 x-2 x^2} \left (1-e^{18+2 e^6+2 e^3 (2-2 x)-6 x+2 x^2}+6 x+4 e^3 x-4 x^2\right ) \, dx=x\,{\mathrm {e}}^{-4\,{\mathrm {e}}^3}\,{\mathrm {e}}^{-2\,{\mathrm {e}}^6}\,{\mathrm {e}}^{6\,x}\,{\mathrm {e}}^{-18}\,{\mathrm {e}}^{-2\,x^2}\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^3}-x \] Input:

int(exp(6*x - 2*exp(6) - 2*x^2 + 2*exp(3)*(2*x - 2) - 18)*(6*x - exp(2*exp 
(6) - 6*x + 2*x^2 - 2*exp(3)*(2*x - 2) + 18) + 4*x*exp(3) - 4*x^2 + 1),x)
 

Output:

x*exp(-4*exp(3))*exp(-2*exp(6))*exp(6*x)*exp(-18)*exp(-2*x^2)*exp(4*x*exp( 
3)) - x
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.35 \[ \int e^{-18-2 e^6-2 e^3 (2-2 x)+6 x-2 x^2} \left (1-e^{18+2 e^6+2 e^3 (2-2 x)-6 x+2 x^2}+6 x+4 e^3 x-4 x^2\right ) \, dx=\frac {x \left (-e^{2 e^{6}+4 e^{3}+2 x^{2}} e^{18}+e^{4 e^{3} x +6 x}\right )}{e^{2 e^{6}+4 e^{3}+2 x^{2}} e^{18}} \] Input:

int((-exp(exp(3)^2+(2-2*x)*exp(3)+x^2-3*x+9)^2+4*x*exp(3)-4*x^2+6*x+1)/exp 
(exp(3)^2+(2-2*x)*exp(3)+x^2-3*x+9)^2,x)
 

Output:

(x*( - e**(2*e**6 + 4*e**3 + 2*x**2)*e**18 + e**(4*e**3*x + 6*x)))/(e**(2* 
e**6 + 4*e**3 + 2*x**2)*e**18)